Resumen Diseño de Maquinas de Shigley cap 2

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Chapter 2

Materials

Lecture Slides

The McGraw-Hill Companies 2012

Chapter Outline

Shigleys Mechanical Engineering Design

Example 1-2


Solution

Answer

Answer

Standard Tensile Test

Used to obtain material characteristics and strengths Loaded in tension with slowly increasing P Load and deflection are recorded


Fig. 21

Stress and Strain


The stress is calculated from

where is the original cross-sectional area.

The normal strain is calculated from

where l0 is the original gauge length and l is the current length corresponding to the current P.

Stress-Strain Diagram

Plot stress vs. normal strain Typically linear relation until

the proportional limit, pl No permanent deformation

until the elastic limit, el Yield strength, Sy , defined at

point where significant plastic deformation begins, or where permanent set reaches a fixed amount, usually 0.2% of the original gauge length

Ultimate strength, Su , defined as the maximum stress on the diagram


Ductile material

Brittle material

Fig. 22

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Elastic Relationship of Stress and Strain

Slope of linear section is Youngs Modulus, or modulus of elasticity, E

Hookes law

E is relatively constant for a given type of material (e.g. steel, copper, aluminum)

See Table A-5 for typical values

Usually independent of heat treatment, carbon content, or alloying


Fig. 22 (a)

True Stress-Strain Diagram Engineering stress-strain diagrams

(commonly used) are based on original area.

Area typically reduces under load, particularly during necking after point u.

True stress is based on actual area corresponding to current P.

True strain is the sum of the incremental elongations divided by the current gauge length at load P.

Note that true stress continually increases all the way to fracture.


True Stress-strain

Engineeringstress-strain

(2-4)

Compression Strength

Compression tests are used to obtain compressive strengths. Buckling and bulging can be problematic. For ductile materials, compressive strengths are usually about

the same as tensile strengths, Suc = Sut . For brittle materials, compressive strengths, Suc , are often

greater than tensile strengths, Sut .


Torsional Strengths

Torsional strengths are found by twisting solid circular bars. Results are plotted as a torque-twist diagram. Shear stresses in the specimen are linear with respect to the radial

location zero at the center and maximum at the outer radius. Maximum shear stress is related to the angle of twist by

q is the angle of twist (in radians) r is the radius of the bar l0 is the gauge length G is the material stiffness property called the shear modulus or

modulus of rigidity.


Torsional Strengths

Maximum shear stress is related to the applied torque by

J is the polar second moment of area of the cross section For round cross section,

Torsional yield strength, Ssy corresponds to the maximum shear stress at the point where the torque-twist diagram becomes significantly non-linear

Modulus of rupture, Ssu corresponds to the torque Tu at the maximum point on the torque-twist diagram


Resilience

Resilience Capacity of a material to absorb energy within its elastic range

Modulus of resilience, uR Energy absorbed per unit

volume without permanent deformation

Equals the area under the stress-strain curve up to the elastic limit

Elastic limit often approximated by yield point


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Resilience Area under curve to yield point gives approximation

If elastic region is linear,

For two materials with the same yield strength, the less stiff material (lower E) has greater resilience


Toughness

Toughness capacity of a material to absorb energy without fracture

Modulus of toughness, uT Energy absorbed per unit volume

without fracture Equals area under the stress-strain

curve up to the fracture point


Toughness

Area under curve up to fracture point

Often estimated graphically from stress-strain data Approximated by using the average of yield and ultimate

strengths and the strain at fracture


Resilience and Toughness

Measures of energy absorbing characteristics of a material Units are energy per unit volume

lbfin/in3 or J/m3 Assumes low strain rates For higher strain rates, use impact methods (See Sec. 2-5)


Statistical Significance of Material Properties

Strength values are obtained from testing many nominally identical specimens

Strength, a material property, is distributional and thus statistical in nature

Example Histographic report for maximum stress of 1000 tensile tests on 1020 steel


Example for Statistical Material Property Histographic report for maximum stress of 1000 tensile tests on

1020 steel

Probability density number of occurrences divided by the total sample number

Histogram of probability density for 1020 steel


Fig. 25

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Example for Statistical Material Property Probability density function (See Ex. 20-4)


Fig. 25

Statistical Quantity

Statistical quantity described by mean, standard deviation, and distribution type

From 1020 steel example: Mean stress = 63.62 kpsi Standard deviation = 2.594 kpsi Distribution is normal Notated as


Strengths from Tables

Property tables often only report a single value for a strength term

Important to check if it is mean, minimum, or some percentile Common to use 99% minimum strength, indicating 99% of the

samples exceed the reported value


Cold Work Cold work Process of plastic

straining below recrystallization temperature in the plastic region of the stress-strain diagram

Loading to point i beyond the yield point, then unloading, causes permanent plastic deformation, p

Reloading to point i behaves elastically all the way to i, with additional elastic strain e


Fig. 26 (a)

Cold Work The yield point is effectively

increased to point i Material is said to have been cold

worked, or strain hardened Material is less ductile (more brittle)

since the plastic zone between yield strength and ultimate strength is reduced

Repeated strain hardening can lead to brittle failure


Fig. 26 (a)

Reduction in Area

Plot load P vs. Area Reduction Reduction in area corresponding to

load Pf at fracture is

R is a measure of ductility Ductility represents the ability of a

material to absorb overloads and to be cold-worked


(2-12)

Fig. 26 (b)

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Cold-work Factor

Cold-work factor W A measure of the quantity of cold work

Shigleys Mechanical Engineering DesignFig. 26 (b)

Equations for Cold-worked Strengths


Example 2-1


Example 2-1 (Continued)


Hardness

Hardness The resistance of a material to penetration by a pointed tool

Two most common hardness-measuring systems Rockwell A, B, and C scales Specified indenters and loads for each scale Hardness numbers are relative

Brinell Hardness number HB is the applied load divided by the

spherical surface area of the indentation


Strength and Hardness

For many materials, relationship between ultimate strength and Brinell hardness number is roughly linear

For steels

For cast iron


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Example 2-2


Impact Properties

Charpy notched-bar test used to determine brittleness and impact strength

Specimen struck by pendulum Energy absorbed, called impact value, is computed from height

of swing after fracture


Effect of Temperature on Impact

Some materials experience a sharp transition from ductile to brittle at a certain temperature


Fig. 27

Effect of Strain Rate on Impact

Average strain rate for stress-strain diagram is 0.001 in/(ins)

Increasing strain rate increases strengths

Due to yield strength approaching ultimate strength, a mild steel could be expected to behave elastically through practically its entire strength range under impact conditions


Fig. 28

Temperature Effects on Strengths

Plot of strength vs. temperature for carbon and alloy steels

As temperature increases above room temperature Sut increase slightly, then

decreases significantly Sy decreases continuously Results in increased

ductility


Fig. 29

Creep

Creep a continuous deformation under load for long periods of time at elevated temperatures

Often exhibits three stages 1st stage: elastic and plastic

deformation; decreasing creep rate due to strain hardening

2nd stage: constant minimum creep rate caused by the annealing effect

3rd stage: considerable reduction in area; increased true stress; higher creep rate leading to fracture


Fig. 210

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Material Numbering Systems

Common numbering systems Society of Automotive Engineers (SAE) American Iron and Steel Institute (AISI) Unified Numbering System (UNS) American Society for Testing and Materials (ASTM) for cast

irons


UNS Numbering System

UNS system established by SAE in 1975 Letter prefix followed by 5 digit number Letter prefix designates material class

G carbon and alloy steel A Aluminum alloy C Copper-based alloy S Stainless or corrosion-resistant steel


UNS for Steels For steel, letter prefix is G First two numbers indicate composition, excluding carbon content

Second pair of numbers indicates carbon content in hundredths of a percent by weight

Fifth number is used for special situations Example: G52986 is chromium alloy with 0.98% carbon


Some Casting Processes

Sand Casting Shell Molding Investment Casting Powder-Metallurgy Process


Hot-working Processes

Process in which metal is formed while heated above recrystallization temperature

Refined grain size Rough surface finish Rolling, forging, extrusion, pressing Common bar cross-sections from hot-rolling


Fig. 211

Cold-working Processes

Forming of metal without elevating temperature

Strain hardens, resulting in increase in yield strength

Increases hardness and ultimate strength, decreases ductility

Produces bright, smooth, reasonably accurate finish

Cold-rolling used to produce wide flats and sheets

Cold-drawing draws a hot-rolled bar through a smaller die


Fig. 212

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Heat Treatment of Steel

Time and temperature controlled processes that modifies material properties

Annealing Heated above critical temperature, held, then slowly cooled Refines grain structure, softens, increases ductility Erases memory of prior operations Normalizing provides partial annealing by adjusting time and

temperature Quenching

Controlled cooling rate prevents full annealing Less pearlite, more martensite and/or bainite Increased strength, hardness, brittleness


Heat Treatment of Steel

Tempering Reheat after quenching to a temperature below the critical

temperature Relieves internal stresses Increases ductility, slight reduction in strength and hardness


Effects of Heat Treating


Fig. 213

Case Hardening

Process to increase hardness on outer surface, while retaining ductility and toughness in the core

Addition of carbon to outer surface by exposure to high carbon solid, liquid, or gas at elevated temperature

Can also achieve case hardening by heat treating only the outer surface, e.g. induction hardening or flame hardening


Alloy Steels

Chromium Nickel Manganese Silicon Molybdenum Vanadium Tungsten


Corrosion-Resistant Steels

Stainless steels Iron-base alloys with at least 12 % chromium Resists many corrosive conditions

Four types of stainless steels Ferritic chromium Austenitic chromium-nickel Martensitic Precipitation-hardenable


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Casting Materials

Gray Cast Iron Ductile and Nodular Cast Iron White Cast Iron Malleable Cast Iron Alloy Cast Iron Cast Steel


Nonferrous Metals

Aluminum Magnesium Titanium Copper-based alloys

Brass with 5 to 15 percent zinc Gilding brass, commercial bronze, red brass

Brass with 20 to 36 percent zinc Low brass, cartridge brass, yellow brass Low-leaded brass, high-leaded brass (engravers brass), free-

cutting brass Admiralty metal Aluminum brass

Brass with 36 to 40 percent zinc Muntz metal, naval brass

Bronze Silcon bronze, phosphor bronze, aluminum bronze, beryllium

bronzeShigleys Mechanical Engineering Design

Plastics

Thermoplastic any plastic that flows or is moldable when heat is applied

Thermoset a plastic for which the polymerization process is finished in a hot molding press where the plastic is liquefied under pressure


Thermoplastic Properties (Table 2-2)


Thermoset Properties (Table 2-3)


Composite Materials

Formed from two or more dissimilar materials, each of which contributes to the final properties

Materials remain distinct from each other at the macroscopic level

Usually amorphous and non-isotropic Often consists of laminates of filler to provide stiffness and

strength and a matrix to hold the material together Common filler types:

Shigleys Mechanical Engineering DesignFig. 214

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Material Families and Classes (Table 2-4)








Youngs Modulus for Various Materials


Fig. 215

Youngs Modulus vs. Density


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Specific Modulus

Specific Modulus ratio of Youngs modulus to density, E /

Also called specific stiffness Useful to minimize weight

with primary design limitation of deflection, stiffness, or natural frequency

Parallel lines representing different values of E / allow comparison of specific modulus between materials


Fig. 216

Minimum Mass Guidelines for Youngs Modulus-Density Plot


Guidelines plot constant values of E/

depends on type of loading

= 1 for axial = 1/2 for

bending

Example, for axial loading,k = AE/l A = kl/Em = Al = (kl/E) l =kl2 /EThus, to minimize mass, maximize E/ ( = 1)

Fig. 216

The Performance Metric


The performance metric depends on (1) the functional requirements, (2) the geometry, and (3) the material properties.

The function is often separable,

f3 (M) is called the material efficiency coefficient.

Maximizing or minimizing f3 (M) allows the material choice to be used to optimize P.

Performance Metric Example

Requirements: light, stiff, end-loaded cantilever beam with circular cross section

Mass m of the beam is chosen as the performance metric to minimize

Stiffness is functional requirement Stiffness is related to material and geometry




From beam deflection table, 3

3FlEI

Sub Eq. (2-26) into Eq. (2-25) and solve for A

The performance metric is

Sub Eq. (2-27) into Eq. (2-28),



Separating into the form of Eq. (2-24),

To minimize m, need to minimize f3 (M), or maximize

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M is called material index

For this example, = Use guidelines parallel

to E1/2/ Increasing M, move up

and to the left Good candidates for this

example are certain woods, composites, and ceramics

Fig. 217



Additional constraints can be added as needed

For example, if it is desired that E > 50 GPa, add horizontal line to limit the solution space

Wood is eliminated as a viable option

Fig. 218

Strength vs. Density


Specific Modulus

Specific Strength ratio of strength to density, S /

Useful to minimize weight with primary design limitation of strength

Parallel lines representing different values of S / allow comparison of specific strength between materials


Fig. 219

Minimum Mass Guidelines for Strength-Density Plot


Guidelines plot constant values of S/

depends on type of loading

= 1 for axial = 2/3 for bending

Example, for axial loading, = F/A = S A = F/Sm = Al = (F/S) lThus, to minimize m, maximize S/ ( = 1)

Fig. 219

Resumen Diseño de Maquinas de Shigley cap 2

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